Static ECDH Curve Selection for Cheon Attack Resistance
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Solution Overview
Problem
The static Diffie-Hellman key exchange in elliptic curve cryptography is vulnerable to the Cheon attack, which makes the protocol less secure due to the ease of solving the static Diffie-Hellman problem when the group size q−1 or q+1 has certain factorizations, compromising the security of cryptographic protocols that reuse secret values.
Innovation Solution
Selecting elliptic curves with specific Cheon-resistant properties, such as q being a prime where q−1=cr and q+1=ds with primes r and s and integer Cheon cofactors c and d such that cd≤48, and using an affine equation y2=x3+ix with a field size p=2454+(3×17×11287)² and order q=2452+(7×41117)², to enhance security against the Cheon attack.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If standard elliptic curve Diffie-Hellman is used, then key exchange efficiency is improved, but security is worsened due to vulnerability to Cheon attack when q-1 or q+1 has certain factorizations
Solution Approach 1:
The patent changes the parameters of the elliptic curve by selecting specific primes r and s, and setting q such that q-1=cr and q+1=ds where c and d are small integers with cd≤48. This parameter selection makes the Cheon attack infeasible while maintaining efficient key exchange operations.
Solution Approach 2:
The patent applies local quality by making specific parts of the cryptographic system (the group order q and its relationship to primes r and s) have special properties that provide Cheon resistance, while other parts of the system remain standard for efficiency.
2Speed
If group size q is reduced for efficiency, then key exchange speed is improved, but security is worsened due to easier solving of static Diffie-Hellman problem
Solution Approach 1:
The patent changes the parameter q to satisfy specific relationships with primes r and s (q-1=cr, q+1=ds with cd≤48), which provides Cheon resistance even at reduced group sizes, thereby maintaining security while enabling faster key exchange.
3Loss of substance
If static private key is reused multiple times, then communication overhead is reduced, but security is worsened due to static Diffie-Hellman vulnerability
Solution Approach 1:
The patent applies preliminary anti-action by pre-selecting elliptic curves with Cheon-resistant parameters (specific q, r, s relationships) before the key exchange protocol is executed. This preliminary selection prevents the Cheon attack from succeeding even when static private keys are reused across multiple communications.
Data Source
AI summary
A method for providing Cheon-resistance security for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the method including providing a system for message communication between a pair of correspondents, a message being exchanged in accordance with ECDH instructions executable on computer processors of the respective correspondents, the ECDH instructions using a curve selected from a plurality of curves, the selecting including choosing a range of curves; selecting, from the range of curves, curves matching a threshold efficiency; excluding, within the selected curves, curves which may include intentional vulnerabilities; and electing, from non-excluded selected curves, a curve with Cheon resistance, the electing comprising a curve from an additive group of order q, wherein q is prime, such that q−1=cr and q+1=ds, where r and s are primes and c and d are integer Cheon cofactors of the group, such that cd≤48.